Measures and LMI for impulsive optimal control with applications to space rendezvous problems

TL;DR

This paper introduces a semi-definite programming approach using measure relaxations and LMIs to find lower bounds and solutions for impulsive optimal control problems, demonstrated on space rendezvous scenarios.

Contribution

It develops a novel SDP-based method for globally solving impulsive nonlinear optimal control problems via measure relaxations and LMI hierarchies.

Findings

Provides lower bounds on the global infimum of impulsive control problems.

Guarantees global optimality through numerical simulations.

Successfully applies the method to space rendezvous problems.

Abstract

This paper shows how to find lower bounds on, and sometimes solve globally, a large class of nonlinear optimal control problems with impulsive controls using semi-definite programming (SDP). This is done by relaxing an optimal control problem into a measure differential problem. The manipulation of the measures by their moments reduces the problem to a convergent series of standard linear matrix inequality (LMI) relaxations. After providing numerous academic examples, we apply the method to the impulsive rendezvous of two orbiting spacecrafts. As the method provides lower bounds on the global infimum, global optimality of the solutions can be guaranteed numerically by a posteriori simulations, and we can recover simultaneously the optimal impulse time and amplitudes by simple linear algebra.